Introducción

Chapter 2
The Hypercube
2.1 Obtaining a Segment, a Square, a Cube and a Hypercube
In [Rucker, 77] is presented the Claude Bragdon's method to define a series of
figures which are called the parallelotopes [Coxeter, 63]. First a 0D point is taken and
moved one unit to the right. The path between the first and the second new point produces a
1D segment. The first dimension, represented by the X-axis, has appeared (Figure 2.1).
X
O
O
FIGURE 2.1
Generation and final 1D unit segment C1 (own elaboration).
The new segment is then moved one unit upward. The path between the first and the
second new segment produces a 2D square (a parallelogram). The second dimension,
represented by the Y-axis, has appeared (Figure 2.2).
Y
O
X
O
X
FIGURE 2.2
Generation and final 2D unit square C2 (own elaboration).
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The new square is then moved one unit forward out this paper. The path between
the first and the second new square produces a 3D cube (a parallelepiped). The third
dimension, represented by the Z-axis, has appeared (Figure 2.3). Because we are working
over a 2D surface (this paper and the computer’s screen), a diagonal between X and Y-axis
represents the Z-axis, however it should be interpreted as a line perpendicular to this 2D
surface.
Y
Y
Z
O
X
O
X
FIGURE 2.3
Generation and final 3D unit cube C3 (own elaboration).
We know that the fourth dimension has a direction perpendicular to the other three
dimensions, in this case the W-axis is presented as a perpendicular line to the Z-axis. Then
the cube is moved one unit in direction of the W-axis. The path (six cubes perpendicular to
the first one) between the first and the second new cube produces the 3D boundary of a 4D
hypercube (a 4D parallelotope). The fourth dimension has appeared (Figure 2.4).
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Y
Y
Z
Z
W
X
O
X
O
FIGURE 2.4
Generation and final 4D unit hypercube C4 (own elaboration).
Definition 2.1: Let Cn be the n-dimensional parallelotope, then C0 is a point and Figures
2.1 to 2.4 correspon...